{
  "abstract": "The classical $A_2$-frame construction of the Coxeter--Todd lattice uses the hexacode as its glue. Here we replace the inert Eisenstein reduction at $2$ by the ramified Gaussian chain ring $S=\\mathbb{Z}[i]/(1+i)^3$. For odd $\\ell$, a one-generator cyclic graph code over $S$ lifts from an $A_2(4)^{2\\ell}$-frame to an even strongly $6$-modular lattice $L_\\ell$ of rank $4\\ell$. In the real frame ordering the same glue is a one-generator negacyclic code. The cases $L_5$ and $L_7$ have minima $6$ and $8$ and give examples for the type-$6\\mathrm{a}$ entries in dimensions $20$ and $28$, which are listed as open in the Nebe--Sloane catalogue. After completing the square, the minimum calculation becomes a four-symbol carry problem; for $L_7$ the key relation is the boundary map of a $7$-cycle. For $\\ell=3$, $L_3$ has an index-$8$ overlattice isometric to the Coxeter--Todd lattice.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27148",
  "author_total": 1,
  "authors": [
    {
      "name": "Ian Teixeira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6c4s2k5m68rkc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27148",
  "fulltext_url": null,
  "id": "6c4s2k5m68rkc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27148",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214178428",
  "pdf_url": "https://arxiv.org/pdf/2609.27148",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6c4s2k5m68rkc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27148",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27148"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$A_2$-frames, negacyclic glue, and extremal strongly $6$-modular lattices",
  "title_text": "A_2-frames, negacyclic glue, and extremal strongly $6$-modular lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/6c4s2k5m68rkc",
  "venue": "arXiv",
  "versions": []
}